A flat metal plate one centimeter thick is shaped like the first quadrant region under the graph of between *x* = 0 and *x* = 4. The density of the plate varies; at a distance *x* units from the y-axis the density is given by x^{2} grams/cm^{3}.

(a) Assume the approximate mass, ΔM, of a strip of width Δx is: ΔM = (volume)(density)

Write a left-hand Riemann sum with 4 terms that approximates the total mass of the metal plate.

(b) Write a definite integral that gives the exact value of the total mass of the plate.

(c) Determine the total mass of the plate.

Quick Find Code For This Problem.

Legend: [QUES2LGND]

Solution For Part a.): [QUES2PA]

Solution For Part b.): [QUES2PB]

Solution For Part c.): [QUES2PC]

Final Answers All Grouped Together: [QUES2ANS]

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[QUES2LGND]

__Legend:__

Important Notes

Useful Info

Answers

Steps Taken

Width = 1 cm

Limits of Integration from x = 0 to x = 4

Δm = (volume)(density)

*In this question, how to find Δm was given. However if it wasn't, we'd use unit analysis to figure it out. ie) The question is looking for mass and the unit for density is given which is grams/cm ^{3}. As soon as you see cm^{3}, you should know that cm^{3} is a measurement of volume. Therefore to end up with a unit of mass we'll need to multiply volume and density. *

**[QUES1PA]**

a) What makes doing these questions easier is to create a visual interpretation of what is going on. Looking at the info given about the flat metal plate, we should be able to create a diagram:

Since Δm = (volume)(density), we will need to find both the values for volume and density respecively. Although, the question has already told us that density = x^{2} grams/cm^{3}. Therefore, all we need to find is the volume of the metal plate. From the diagram above we can see that volume = (length)(width)(height). Filling in the the values we get:

Now that we have both the density and the volume, we can now substitute in the two values and find out what Δm looks like.

Since the question asks for a left handed Riemann sum we **first start with x = 0 and increase by 1 unit until we reach x = 3 to make the four equal subintervals**. Riemann sums take the sums of the areas of the rectangles made under the curve with the lengths of said rectangles being the value of the function and the widths being the change in x. So your Riemann sum should look something like this:

The answser is 49/12 grams!

**[QUES2PB]**

b) Unlike in Part A, where we found an approximation of the mass, due to using a Riemann sum with subintervals of 1 unit wide, we will use integration to solve for the total mass. The reason why integration is more accurate than the Riemann sum is because of the fact that integration takes infinitesimally Δx, the interval widths of the rectangle, therefore causing a more accurate representation of what we are trying to calculate, in this case, the mass. Also, since Δx gets infinitesimally smaller, it eventually turns into what we call dx.

So an integral solving for the exact answer for the total mass of the plate is:

**[QUES2PC]**

c) Now here's the hardest part of the question. Our task is to solve for the integral found in Part B but since we're not allowed to use a calculator, we'll be antidifferentiating by hand. I first used integration by parts but got stuck when I had to antidifferentiate ln(1+x). That lead me to my next method, which was using substitution. It turned out much much nicer.

I'm going to **substitute u for 1+x.** Now finding the derivative of u, we see that du = dx. **I also need to rewrite the limits in terms of u instead of x**.

__Lower Limit:__

u = 1 + x

= 1 + 0

= 1

__Upper Limit:__

u = 1 + x

= 1 + 4

= 5

Writing everything out, we end up with:

It's always easier to work with polynomials so we should expand the numerator, and simplify!

In this step I found the antiderivative of each term, as stated by the Additive Interval Rule. Then I plugged in my limits. It's all arithmetic after that.

So the total mass of the plate is exactly 4+ln5 grams.

*The Riemann sum was a bit off. It was an underestimate. That gives us a little idea about how the graph of Δm looks. Since the left hand estimate was lower than the actual answer, the function should be increasing at [0,4].*

**[QUES2ANS]**

__Concluding Answers:__

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